Determination of the Minimum-Variance Unbiased Estimator for DC Power-Flow Estimation

Determination of the Minimum-Variance Unbiased Estimator for DC Power-Flow Estimation Mohammadhadi Amini, Arif I. Sarwat, S. S. Iyengar, Ismail Guvenc Department of Electrical and Computer Engineering, College of Engineering and Computing Florida International University Miami, Florida 33174, USA hadi.amini@ieee.org; asarwat@fiu.edu Abstract One of the most important features of the Smart Grid (SG) is real-time self-assessment which may threat that target power system stability. In order to improve robustness of power systems against such attacks, accurate estimation of the power system operation is required and conventional power flow methods should be upgraded. In this paper, we derive minimum variance unbiased estimators (MVUEs) for active power based on the voltage phase at each node of the power system. The state variables are the voltage phases and the received measurement signals are active power measurements. The proposed method is implemented on a four-bus test system. Three scenarios are defined to investigate the effect of covariance matrix topology on the estimation accuracy. The results shows that lower correlation between the noise vector elements leads to a more accurate estimation of power system operation. Keywords DC power-flow, Minimum-Variance Unbiased Estimator (MVUE), State Estimation. I. INTRODUCTION Considering the growing awareness about energy and environment, the demand for a reliable and sustainable power grid and the need for high quality resources leads to the evolution of Smart Grid (SG) as a novel means to worldwide electricity grid [1]. There are many effective elements in the future power system, such as transportation electrification [2], [3], distributed renewable resources [4], smart appliances [5] and hybrid DC migrogrids[6]. Furthermore, SG will gradually lead to the development and upgrading the whole power grid, that creates a more secure, energy-saving, environmental-friendly, and economic power system. Selfhealing and self decision making techniques for SG systems pave the way towards an adaptive and intricate future power system [7], [8], [9]. For example, the approach in [9] utilizes multi-agent systems in order to implement an autonomous load management framework. In [10], an energy management system is proposed for the optimal operation of SG and microgrids. This approach is based on hybrid connected neuron networks and optimal power flow. Notwithstanding, there is an urgent need to introduce more accurate and fast power flow methods, which is the main focus of the present paper. Power-flow studies are performed to determine the steadystate operation of an electric power system. They target calculation of the voltage drop on each feeder, the voltage at each bus, and the power flow in all branch and feeder circuits. Contingencies (fault, congestion, physical attack, generator failure, etc) are important issues in the power system operation. Furthermore, the contingency analysis plays a pivotal role in the future power system. For instance, in the conventional power system the operating personnel need to realize what power-flow changes will occur due to particular equipment outage. The real-time contingency data can be utilized in order to forecast problems caused by such outages, and can be used to develop operating schedules in order to overcome the problem. In the SG, there is a crucial need to enable the self-decision making ability. Any inaccuracy in the power-flow calculation may lead to a major outage or black-out [11]. The main objective of utilizing power-flow method is to calculate the amount of active power, reactive power, and voltage phase in the power system. Four well-known methods in the power-flow analysis can be summarized as follows [12], [13]: 1) Gauss-Seidal method which is an iterative approach for solving linear systems. This method has the ability to be applied to any matrix which its determinant is nonzero. The matrix should be diagonally dominant, or symmetric and positive definite so that the convergence of method is guaranteed. 2) Newton Raphson method is one of the powerful techniques. Generally, the convergence is quadratic: as the method converges on the root, the difference between the root and the approximation is squared (the number of accurate digits roughly doubles) at each step. However, there are some difficulties with the method. 3) Fast Decoupled method:fast power-flows algorithms are utilized to propose power-flow solutions in short durations (seconds or a proportion of seconds). 4) DC power-flow method: Further simplifications on the fast power-flow algorithms is achieved by expanding the fast decoupled power-flow to neglect the Q V equation, assuming the voltage magnitudes are constant at 1.0 per-unit [14]. This simplification and the equation related to the DC power-flow will be discussed in the problem definition. In [15], some of the advantages of DC power flow are mentioned as follow: non-iterative, reliable and unique solutions, simple methods of solving, and reasonable accuracy of approximated MW flows. Some studies tried to analyze the theoretical error of DC models in power flow [16], [17]. Bidirectional energy transfer in DC microgrids is one of the motivations to focus on DC power-flow method [18]. Hybrid AC/DC microgrids control requires DC power-flow method for studying the technical aspects [19]. The main objective of this paper is to utilize the state estimation approach for linear problems in order to estimate the DC

{ H H αβ = αβ = N i=1,i =α B αi ; α = β B αβ ; otherwise. (7) Fig. 1. Power-flow between two buses. power-flow. The contribution of this paper, in comparison with past researches, is to introduce an estimator for DC powerflow based on MVUE. The rest of the paper is organized as follows: section II defines the problem elaborately. In section III the linear model is introduced in two specific parts: 1) white Gaussian noise, and 2) generalized Gaussian noise. Section IV is about the numerical study on the test system, which is a four bus power network. Three scenarios are defined in this section to evaluate the effect of covariance matrix on the estimation accuracy. Finally, section V is the conclusion. II. PROBLEM DEFINITION In this paper, we use the DC power flow method in order to calculate the active power based on the voltage phase of each node. As it is mentioned in the introduction, the nature of DC power flow problem is linear. According to some mentioned simplification in the inroduction section, the power-flow on the line from bus j to bus k with reactance X jk becomes: P jk = δ j δ k X jk, (1) where δ i shows the voltage phase at the i th bus, X jk is the reactance between the j th bus and the k th bus, and P jk represents the active power flow between the j th bus and the k th bus. Hence,the real power balance equations reduce to the following linear equation: Bδ = P, (2) where B is the imaginary component of the Y bus calculated neglecting line resistance and excepting the slack bus row and column, and and P represents the vector containing theactive power flows.. With the advent of power system restructuring, the DC power-flow has become a commonly used analysis technique. The following equation illustrates the linearized form of power flow considering the measurement errors: x = Hδ + e, (3) where x is a vector which contains the active power measurements, H is a square matrix which is calculated based on the power system topology and inductance values. We can show x, H, δ and e as follows: x =[ P 1 P 2... P n ] T, (4) δ =[ δ 1 δ 2... δ n ] T, (5) e N(0, R), (6) In order to go through the next step and obtain the estimator, we considered that the noise is colored Gaussian, and the joint probability density function of x is given by: { 1 1 p(x; δ) = exp (2π) n/2 1/2 R 2 (x Hδ)T R 1 (x Hδ), (8) where R =det(r), and det(.) denotes the determinant of a matrix. As a corollary, the problem is defined in this section. The next step is to introduce the general linear model for finding the minimum-variance unbiased estimator (MVUE) for the proposed problem. III. LINEAR MODEL AND DERIVATION OF MVUE In this section, first, the linear models will be introduced for a specific noise vecttor. Then, the general linear model is discussed [20]. Both of the introduced models are important and will be assessed in the simulation section. A. Linear Model Considering a noise vector with PDF N (0,σ 2 I). ˆδ = g(x) is an MVUE if: ln p(x; δ) = I(δ)(g(x) δ), (9) with Cˆδ = I 1 (δ). Therefore, we need to factor the derivative of the natural logarithm of (8) into the form I(δ)(g(x) δ) in order to find the MVUE. ln p(x; δ) = [ ln(2πσ 2 ) N/2 1 } ] 2σ 2 (x Hδ)T (x Hδ). (10) After some manipulations, the MVUE for δ is as shown in the following equation: ˆδ =(H T R 1 H) 1 H T R 1 x, (11) which has a covariance matrix given by Cˆδ = σ 2 (H T H) 1. B. Generalized Linear Model In the generalized linear model, two important extensions have been done in order to achieve a more generalized model that can be used for general Gaussian noise. The x vector may now include a known signal term,s, i.e. [20]: x = Hδ + s + e, (12) where s is a N 1 vector of known signal samples and e is the noise vector with general Gaussian PDF N (0, R). In order to facilitate the modeling process, we can use the whitening transformation and change the problem to the linear model which is discussed in the earlier part. If we factor the noise covariance matrix, R, as: R 1 = D T D, (13)

B. Linear Model Parameters for the Test System In order to find an MVUE of δ, we need to find the matrix H. The noise vector should be defined on a scenario-based approach to provide us the ability to compare these scenarios and the effect of covariance matrix of the noise, R on the estimation accuracy. If we want to use the admittance values shown in Fig. 2 to achieve the H matrix, we should use (5). The important point is that B αβ = Y αβ. The H matrix can be written as follows: 30 10 10 10 10 30 10 10 H = 10 10 30 10 (17) 10 10 10 30 Fig. 2. Four bus test system. then we can use the matrix D for the whitening transformation. If e = De, the PDF of e is N (0, I). After this transformation we have: x = Dx = DHδ + Ds + De = H δ + s + e. (14) Now, we define x = x s.sowehave: x = H δ + w The MVUE of δ for the given observed data x is: ˆδ =(H T H ) 1 H T x =(H T D T DH) 1 H T D T D(x s). (15) By replacing D T D with R 1,wehave: ˆδ =(H T R 1 H) 1 H T R 1 (x s), (16) and the covariance matrix is shown by Cˆδ =(H T R 1 H) 1. Now, the problem is defined and the tools for solving linear model and calculate the MVUE for a linear model for the specific noise (white Gaussian) and general noise (general Gaussian) is discussed elaborately. In the next section, the case study (four bus test system) and its specifications will be introduced and then the mentioned modeling will be implemented on this system to analyze the different scenarios. IV. NUMERICAL STUDY This section includes three parts: 1) introducing four-bus test system, 2) extracting the required matrices and vectors for the modeling purposes, 3) obtaining the MVUE for three specific scenarios, and 4) comparing the results of estimation for three defined scenarios. A. Four-Bus Test System Fig. 2. shows the test system which is used for the study of the linear model. As this figure shows, there are three generators at the 1 st, 2 nd and 4 th bus with the values of 2 p.u, 2 p.u and 1 p.u, respectively. Two loads are located at the 2 nd and 3 rd bus with the value of 1 p.u and 4 p.u, respectively. The admittance between the i th and j th bus is denoted by y ij. Vector δ which is calculated using the mentioned H matrix for the case study is: δ =[ 0 0.025 0.15 0.025 ] (18) C. Obtaining the MVUE for the Test System The three scenarios are designed to show the difference of white Gaussian noise and correlated noise. Therefore, three different covariance matrices are considered. These scenarios are a sample of real power system. The first scenario illustrates a power grid in which the topology is enough robust that the noise of any measurement in any bus, does not affect the noise in another bus. The second scenario is proposed to evaluate the effect of noise variance on the estimation accuracy. Finally, the third scenario is related to an intricate power system in which the noise of each measurement at any bus affected by the other buses noise. First Scenario: In this scenario we have: R = N (0,σ 2 I). (19) where σ 2 =0.2 is assumed for the computation purposes. The observation vector, x, for this scenario is shown in the following equation: x 1 =[ 2.0160 0.5853 3.9177 1.1354 ] (20) The results of estimation for δ is: ˆδ 1 =[ 0.0019 0.0293 0.1464 0.0201 ] (21) The calculated x based on the estimated δ is: ˆx 1 =[ 2.0320 0.3656 3.8354 1.2708 ] (22) The results of this scenario are shown in Fig. 3. Second Scenario: In this scenario we have: R =diag(σ 2 1,σ 2 2,σ 2 3,σ 2 4), (23) where the value of σ 2 1, σ 2 2, σ 2 3 and σ 2 4 are 0.2, 0.4, 0.1, and 0.25, respectively. The results of this scenario are shown in Fig. 4.

Fig. 3. Comparison between real and estimated x matrix elements, first Fig. 6. EEP vs noise variance for the second EEP 1 =[ 0.7937 37.5363 2.1007 11.9253 ] (26) EEP 2 =[ 2.9266 17.1016 2.2722 4.6244 ] (27) EEP 3 =[ 23.7810 110.3894 14.2768 38.3468 ] (28) Fig. 4. Comparison between real and estimated x matrix elements, second Third Scenario: In this scenario, we have the general Gaussian noise which means that there is correlation between noise vector elements. The covariance matrix for this scenario is as below: 0.2 0.04 0.04 0.06 0.04 0.08 0.072 0.048 R = 0.04 0.072 0.4 0.056 (24) 0.06 0.048 0.056 0.4 The results of this scenario are shown in Fig. 5. In order to compare the results of these scenarios, the estimation error is calculated. Figures 2, 3 and 4 illustrate the difference between real and estimates x vector for the first, second and third scenario, respectively. In order to investigate the effect of covariance matrix on the estimation accuracy, the Estimation Error Percentage (EEP) defined as below: x(i) ˆx(i) EEP = 100. (25) x(i) The value of this error has been calculated for three scenarios and shown as follows (values are in percent): Fig. 5. Comparison between real and estimated x matrix elements, third Fig. 6 illustrates the value of estimation error versus the variance of the noise. This figure shows that by increasing the variance, the accuracy of the estimation will decrease. V. CONCLUSION In this paper, a novel power system operation analysis method is introduced. This study can be utilized in the future power system in order to analyze the received data from modern measurement units, such as Phasor Measurement Units (PMUs), smart meters, and Intelligent Electronic Devices (IEDs). In the proposed approach, first, the DC power-flow method is used to calculate the power-flows based on the voltage phases. Then, based on the linear model of DC powerflow, the MVUE is attained. The power-flow estimation is implemented based on the MVUE. In order to investigate the effect of noise covariance matrix on the accuracy of estimation process, three different scenarios were defined. The results show that higher sparsity in the covariance matrix (less correlation between noise vector elements) leads to better accuracy. Furthermore, the case study shows that in the case in which the noise is uncorrelated, the estimation results are more accurate than the case in which the noise is correlated. REFERENCES [1] V. C. Gungor, D. Sahin, T. Kocak, S. Ergut, C. Buccella, C. Cecati, and G.P. Hancke, Smart grid technologies: communication technologies and standards, IEEE Transactions on Industrial Informatics, vol. 7, no. 4, pp. 529-539, Nov. 2011. [2] W. Su, H. R. Eichi, W. Zeng, and M.-Y. Chow, A Survey on the Electrification of Transportation in a Smart Grid Environment, IEEE Transactions on Industrial Informatics, vol. 8, no. 1, pp. 1-10, Jan. 2012. [3] M.H. Amini, and A. Islam Allocation of electric vehicles parking lots in distribution network, in Proc. IEEE Innovative Smart Grid Technologies Conference (ISGT), pp. 1 5, Washington, DC, USA, Feb. 2014. [4] J. M. Carrasco et al, Power-Electronic Systems for the Grid Integration of Renewable Energy Sources: A Survey, IEEE Transactions on Industrial Electronics, vol. 53, no. 4, pp. 1002-1016, Jun. 2006. [5] Xinghuo Yu, C. Cecati, T. Dilon, and M. G. Simoes, The New Frontier of Smart Grids, IEEE Industrial Electronics Magazine, vol. 5, no. 3, pp. 49-63, Sep. 2011.

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